If virtual particles have negative mass why do they contribute positive mass to atoms?

According to Lawrence Krauss, atoms containing in our body consists of merely 10% (if I remember correctly) of our total mass. The rest come from virtual particles popping in and out of existence from empty space. So if ~90% of our mass comes from virtual particles, and virtual particles have negative mass, shouldn't virtual particles mass be subtracted to ours? In other words, why we aren't getting lighter due to the effect of those particles?

• Virtual particles as a model for the binding energy of the quantum fields in protons and neutrons don't have negative mass. Instead they're "off shell" so they have a different mass than a particle in that field should. – Brandon Enright Aug 1 '14 at 4:46
• – Brandon Enright Aug 1 '14 at 4:47
• You'll also find profmattstrassler.com/articles-and-posts/… useful. – Brandon Enright Aug 1 '14 at 4:48
• @BrandonEnright very helpful, I've got something to scratch my head now. Thank you very much :). Also could you post those comments as answer so I can accept it? :) – Davita Aug 1 '14 at 4:50
• Dear @BrandonEnright ,they are off-shell in general but they are surely allowed to have negative energy, too. – Luboš Motl Aug 1 '14 at 6:01

The underlying level of nature is quantum mechanical and obeys special relativity laws, not Newtonian mechanics. Mass is a conserved quantity in Newtonian mechanics, but not in the underlying quantum mechanical framework.

The classical framework emerges as the variables become large enough where h_bar, which characterizes the quantum mechanical level, can be safely assumed zero, because it is so small with respect to the constants entering classical equations.

In the quantum mechanical framework the basic building blocks are the elementary particles that have very small invariant mass. The invariant mass is the "length" of the corresponding four dimensional vector that describes a system. In a similar way that in three dimensional space, if one adds vectors the length of the resultant vector is not a linear addition of the length of the vectors but can be much smaller than the sum, in four dimensional vector additions the invariant mass is controlled by the four dimensional vector algebra of the pseudoeuclidean space that is the basis of special relativity.

The masses of the quarks entering the proton when summed have mass less than 15 MeV, and the proton has a mass close to 1000. The mass of the proton comes up by the vector addition of the four dimensional vectors entering the problem, gluons, quark antiquark pairs that are holding the three basic quarks bound as a proton. These particles are called virtual because they cannot be extracted and measured, they are a part of the necessary calculations. They are characterized by the quantum numbers of the named particles, but are off mass shell. They follow the four dimensional algebra of special relativity. The invariant mass resulting by adding up these innumerable constituents' four vectors, gives the mass of the proton.

As we go to larger space dimensions, the dimensions of nuclei, the difference between adding the constituent nucleons (protons and neutrons) masses and the mass of the nucleus they compose is much smaller, but still exists, and we see it as the binding energy of the nuclei. This is what has been utilized in getting nuclear energy in reactors and bombs.

At the sizes of crystals and solid state lattices the energy differences between adding constituent masses and the mass of the solid is even smaller since the levels are now in electron volts, and is responsible for the chemistry and solid state of matter.

At the level of our every day life, Newtonian physics holds and we can assume masses are conserved, unless there is a nuclear interaction. The accuracy in measurement should be quite good to see the differences due to chemical interactions.